Positive AlgebraFrom arithmetic to

algebra

Jaap den Hertog

Freudenthal Instituut

Universiteit Utrecht

J.denhertog@fi.uu.nl

“I used to be good at arithemetic, but now I don’t understand anything anymore.”

Counting in primary school grows into advanced and more sophisticated counting

You cannot maintain what you never learned

When do you use your calculator?

Continuous learning trajectories

To introduce negative numbers and to use them

Knowledge about fractions as a preparation to working with algebraic expressions

Rules, patterns, structures

27 – 38 = ….?

5 × -3 = -15 -1 × -3 = 3

4 × -3 = -12 -2 × -3 = 6

3 × -3 = -9 always 3 more

2 × -3 = -6

1 × -3 = -3

0 × -3 = 0

A pattern

What is the power of algebra?Reasoning and generalizing: is it always?

Are you sure? Is it certain?

Not only knowledge of (f.e. number system) but also knowledge about

Development of thinking models

A continous learning trajectoryDeveloping a fraction languageReasoned dividePerform operations within the contextTo relate ‘Part of’ to multiplicationTowards the development of routine

proceduresFractions on the number lineAnd what is next …?

Two thirds of 4500

2/3 times 4500

× 45002

3

A learning process and struggles

π/4; 1/4π; π ÷ 4; they are all the same, but differentAdd up the same number with the nominator and

the denonminatorYou divide a number and the result is larger. Why?Add up the nominators and the denominators. Is the

new fraction bigger or smaller than the sum of the fractions?

Is there a smallest fraction greater than zero?How is the number system extended?

15---

35---

25---

45---

A square of 1 bij1. Write the area of each piece as a fraction and add up.

When is formal arithmetic with letter fractions introduced?

For which students is it important?

In which grade do we start?

What are the preparations for the students?

Which formula is equivalent with…

2 3

1y

x x

3 1

2 53 5

2 15( 3)

2( 1)

x x

x

xx

x

Are there more examples?Is there a formula?

1 14 1 4 1

3 3

Simplify fractions

1

2

a

a

Reasoning with formulas

Adjust / prepare formulas yourselfDiscus the effect of changes in variables and /

or numbers

Recommended maximum heart rate

For years, the following formula was used:

Maximum heart rate = 220 – age

Who has a higher maximum heart rate, someone in your class or one of the teachers?

Recommended heart rate

Recently the formula has been changed

Maximum heart rate = 208 - (0.7 x age)

What are the consequences of using this formula: is your heart rate higher or lower than the recommended rate?

Summary

Continuous learning trajectories from Primary school and Secondary school

Introducing negative numbers in primary school, but the formal operations in secondary school

Fractions are not “ready” after the primary schoolFractions in secondary school Do not avoid fractions in secondary education, but

also include lettersLearning processes in developing and adapting

formulas